It is a fairly old term, I have heard it used since the mid 1960's. This is just a guess, but I would say that it comes from the steel refining process in use in the 1960's and earlier, mainly the open hearth process. In an open hearth furnace, after the bath is up to temperature, iron ore is added to provide a source of oxygen to burn out the carbon from the pig iron, which the was the initial molten metal charge to the furnace. The oxygen causes the bath to boil. The longer it boils, the more carbon is burned out. The boil would be vigorous/agitated at the start and diminish until it became "mild" near the end. The (low carbon) steel produced when the boil had become mild would thus be termed "mild steel".
I did some searching, and didn't find an origin- but I think it is much older than 1960's, say, early 1900's or earlier. One of the definitions was simply steel that could not be hardened.
